H.K. Moffatt- On the suppression of turbulence by a uniform magnetic field

8/3/2019 H.K. Moffatt- On the suppression of turbulence by a uniform magnetic field

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J . Flu id M ech . (1967) , vol. 28 , part 3, p p . 571-592Printed in Oreat Britai n

571

On the suppression of turbulence by auniform magnetic fieldBy H. K. MOFFATT

Department of Applied Mathematics and Theoretical Physics, Silver Street,Cambridge(Received 28 September 1966)

The suppression of initially isotropic turbulence by the sudden application of auniform magnetic field is considered. The problem is characterized by threedimensionless numbers, a Reynolds number R, a magnetic Reynolds number R,and a magnetic interaction parameter N (for definitions, see equations ( 1 . 2 ) and(1.4)).It is supposed that R B 1 , R, < 1 and N 9 1. There are two importanttime scales, t , a time characteristic of magnetic suppression, and to = Nt,, theturn-over time of the turbulent energy-containingeddies. For 0 c < to the

response of the energy-containing components of the turbulence to the appliedfield is linear and the time dependence of the kinetic energy density K(t )andmagnetic energy density M ( t ) are analysed. There are essentially two distinctcontributions to each from two domains of wave-number space g1andB2 definedin figure 2). I n gl he response is severely anisotropic, while in B2 t is nearlyisotropic. The relative importance of the contributions K l ( t ) from Bl)and K 2 ( t )(fromB2)o K(t ) epends on the value of the Lundquist number S = (NR,)B. IfS < 1 , then K,(t ) dominates for all t 5 to and K ( t )cc t-3 for t , < t < to . IfS 9 R k 2 , hen K 2 ( t ) ominates, and K(t) c t-8 for R,t, < t < to . If 1< S < R;2,then a changeover in the dominant contribution occurs when t = O(S3R,)t0.Analogous results are obtained for the magnetic energy density.

1. IntroductionIt has been pointed out by Lehnert (1955) that a uniform applied magnetic

field B, has a pronounced effect on the decay of turbulence in a conducting fluid,If the kinematic viscosity of the fluid v is small compared with its magneticdiffusivity A, and if circumstances are such that non-linear terms in the govern-ing equations are negligible, then the effect of the field is to suppress preferentiallythose Fourier components of the velocity field whose wave vector k has a non-zero component parallel to B,, i.e. which tend to bend the field lines. Fouriercomponents for which k .B, = 0 are unaffected by the field-an effect which isalso well known and of profound importance in the related context of stabilitytheory. The time characteristic of this preferential damping process is

t , = h/hi, (1.1)where h, = (,up)-BB,,U is the magnetic permeability and p the density (assumeduniform) of the fluid.

http://moffatt.tc


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572 H . K . MofjfuttA second effect of the magnetic field is that a certain range of Fourier compon-

ents of the velocity field of small wave-number tend to propagate as Alfvdnwaves. If the dominant contribution to the energy of the turbulence comes fromsuch components, then the turbulence will have the character of a random sea ofinteracting Alfvh waves. In particular, during the last stages of decay of homo-geneous turbulence in the presence of a uniform field, when the length scale of theenergy-containing eddies increases without limit, the turbulence must have thischaracter. The final period of decay has been the subject of several papers,following Lehnert (1955) ;the relation of these contributions to the present s tudywill be established in the appendix.

The specific situation studied in this paper is the following. Suppose th at fort < 0 a conducting fluid is in a s tate of turbulent motion characterized by a root-mean-square velocity U,, and a length scale (of energy-containing eddies) I , , with

The time scale of the energy-containing eddies (the turnover time) isto= zo/uo* ( 1 . 3 )

Suppose that , at the instant t = 0, a uniform magnetic field B, is switched on(whether this is a physical possibility is a separate matter which will be discussedbelow), and suppose that

(1.4)E - = -o W O 1 ,t d huoi.e. t , < to . N is usually described as the magnetic interaction parameter and mayalso be interpreted (when R, < 1) as the ratio of the magnitude of the Lorentzforce I(V A h)A h,l to the magnitude of the non-linear part of the inertia force/U. ul in the equation of motion.? The preferential damping of the turbulencethen proceeds on a time scale short compared with the time to characteristic ofthe energy transfer associated with the non-linear forces. For 0 < t < to , theturbulence rapidly adjusts to the new externally applied conditions, and duringthis period of adjustment$ the non-linear forces are negligible (as are viscousforces U fortiori), and the linear decay equations studied by Lehnert are applic-able. For t = O(t,) and greater, non-linear forces may again become important indetermining the new spectral distribution of energy.It is well known in other contexts (Batchelor 1953, chap. IV) that , if externalconditions are suddenly changed, then turbulence will respond in a linear manner

If N < 1, then the Lorentz force is small compared with the inertia force, and may beexpected t o have negligible effect on the turbulence. In th is case, the turbulence generatesweak electric currents whose spectrum is related in a simple way to the spectrum of thevelocity field (see Liepmann 1952; Golitsyn 1960; or the review article by Moffatt 1962).$ This statement requires refinement when the details of the non-linear process areanalysed. It is possible that non-linear effects become important after a time of ordert, = N Y t , where 0 < y < 1. The value of y depends on the detailed nature of the non-linear process, which has not yet been satisfactorily analysed. I n any case, it is certain tha tt , 9 t , if N is sufficiently large and linear theory is valid until t = O(t,). We shall use thepreliminary crude estimate y = 1 until analysis definitely reveals an inconsistency.


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Suppression of turbulence b y a magnetic f ield 573during a time small compared with the time to.The damping of turbulence byconvection through a wire gauze, and the distortion of turbulence by passagethrough a duct of rapidly varying cross-section have been analysed by lineartechniques. I n both cases the theory attempts only t o relate the statistical pro-perties of the turbulence immediately after the change of conditions (i.e. fort < to ) o it s properties immediately before.The feasibility of switching on a field B, at the instant t = 0 requires somecomment. If we imagine that the turbulent fluid is contained in a cubical box ofside L, B I,, and if the source of the field is an electromagnet with coils surround-ing the box, then it takes a finite time of order t, = from the moment ofapplying power t o the electromagnet, for the field to penetrate to the centre ofthe box; t, may be described as the switch-on time for the field, and, if the process

UFIGURE .The sudden application of a magnetic field; grid turbulence is convected into theregion of uniform field between magnet poles N, S.is to be regarded as instantaneous, t, must be small compared with t,, and clearlythis condition can be satisfied only if

S h,l,/A = (NR,)* < lo/Lo. ( 1 . 5 )X s the Lundquist number, and it will play a central role in the analysis of thispaper. This suggests that, for strictly homogeneous turbulence (L, CO), theswitch-on specification of the problem is legitimate only in the limit S-tO;alternatively, one could say that the switching on of the field will introduce in-homogeneities on the scale AS-~Z,;provided X < 1, it is legitimate to assume tha thomogeneity persists, a t any rate on the scale I , of the energy-containing eddies.

If S < 1 , the simplest way to realize the above conditions in the laboratorymight be simply t o sweep grid turbulence into a region of uniform field (figure 1).The switch-on time is then t, = d /U , where U is the uniform mean velocity and dthe fringeing distance of the field, i.e. approximately the gap width of the magnetN-S. The conditions t , < t , < to are then satisfied provided

U h2 U- -B B-0 ,d h 1,and, since the ratio U,/ U can be made small simply by decreasing the dynamicalresistance of the grid used, it should be possible to satisfy these conditions. (Of


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574 H . K . Mofjattcourse, the magnetic Reynolds number based on U , U d / h ,must remain small,since, otherwise, the field of the magnet would be significantly distorted.)1, then strong inhomogeneity must develop if a field is switched on asdescribed above. For turbulence in a box, one would expect the eddies near theboundaries to be first affected by the inward-diffusing field. The problems raisedare certainly interesting, but mathematically intractable. An alternative specifi-cation of the problem which does not exclude homogeneity is as follows. Supposethat,for t < 0, the fluid is at rest and permeated by a uniform field B,, and supposethat, a t the instant t = 0, a homogeneous random velocity field is generated byrandom impulsive forces (Saffman 1967) . I n principle, any initial velocity fieldu(x,0) may be generated by an impulsive force distribution p-lu(x,0)8(t); n theabsence of any impulsive electromotive forces , magnetic field perturbationsh(x, ) take a finite time to develop, so that h(x,0) = 0. The problem then is t oinvestigate how u(x, ) evolves statistically, particularly during the initial phase 0 < t < to. It is by no means obvious whether such conditions can be realized,even approximately, under laboratory conditions. Nevertheless, the problem isof considerable fundamental interest; it is essential to understand fully how afield of turbulence is initially modified by a uniform magnetic field, before therecan be any hope of understanding fully non-linear effects.

When R, < 1 , the induced field h(x, ) is of order R,h, and satisfies the linear-ized induction equation (Lehnert 1955)

If S

ah/at= h, .VU +hV2h. (1 .6)The relevant initial condition for the problem studied here is h(x,0) 0. Thevelocity field u(x, ) is determined by the equation of motion, and for 0 < t < toas explained above (assumingN B 1 ) it takes the linearized inviscid form

au/at= -p-lVP+h,.Vh, (1.7)where P is the sum of the fluid pressure and the magnetic pressure. The initialconditions here are that all statistical properties of the field u(x,0) are supposedgiven. I n order to carry out the limited analysis of this paper, only the spectrumtensor of the field u(x,0) will be required; a discussion of the most appropriateinitial conditions will be deferred to 5 3. The fields U and h also satisfy the kine-matical conditions V . U = V.h = 0. (1.8)

One of the aims of this paper will be to determine the time dependence of suchquantities as the kinetic energy density during such time intervals as that definedby t, < t < to. Although the physical meaning of such an interval may be clear,its mathematical definition may require a word of comment. Two limiting pro-cesses are considered, t,/t, = N +m and t -+m . If the two processes are ied, insuch a way tha t t = o ( N ) ,e.g. t = O ( N t ) , hen the two limits

t/t,+m, t / t , +O ,$(t , N ) f ( t ) ( t d < t


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Suppressionof turbulence by a magnetic ield 575will be used to mean the same as

Similarly, for example, the notation@ ( t , ~ )o(t-3) ( td< t < Nt,)

will mean lim t3@(t,N)= const.t+m,N-mt = o ( N )

In order to avoid tedious repetition, we shall often simply write t % t, and omitthe explicit dependence on N , but it must be stressed that the analysis of $ 3 isliable to break down when t = O(Nt,).Some of the statistical properties of the solutions of Lehnerts equations havebeen numerically computed by Deissler (1963) . Those of his results that arerelevant to the present study will be discussed in 0 3.A preliminary calculation of the initial response of turbulence to the applica-tion of a strong magnetic field, by means of a Taylor series in time, has beencarried out by Nestlerode & Lumley (1963) . This provides an indication of theresponse while t < t,, when the suppression effect is still very slight. The presentpaper goes very much further, by clarifying the nature of the suppression processwhen t 9 t, and the turbulence has been very considerably modified.

2. The suppression of a single Fourier componentEquations (1 .6 ) and (1 .7 ) are well known as the equations that describe decay-ing Alfv6n waves. Certain properties of the solutions of the equations that are ofcentral importance in later sections are recapitulated here. Let us restrict atten-tion to fields which admit a Fourier representation of the form

I(x,t)= p(k,t)exp(ik.x)dk,sP(x,t)= s II(k,t)exp(ik.x)dk,)

where k = (kl,k2,k3), k = dk,dk,dk,, and the k3-axis is chosen parallel to h,.Then the Fourier transforms of equations (1 .6 ) , (1 .7 ) and (1.8)are

(2 . 2 )ip/at = -ikII/p+i(h,.k)q,

[ (a /a t )+Ak2]q = i(h,.k)p,k . p = k . q = 0.

It follows immediately that k211 = 0, and hence that kII E 0. Equations ( 2 . 2 )then admit decaying solutions, p, q K e-pt, where ,L3 is a root of the quadraticequation -p( - p + A k 2 ) + (ho.k)2= 0,i.e. /3 = frAk2[ 1i 1- 6741 = pl, 2 , say; (2 . 3 )


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576 H . K . Moflatthere 6 is the parameter introduced by Lehnert (1955)

6= 2h0.k / k 2 h= ( 2 k d / k ) OS 0, (2.4)where kd, he 'dissipation wave-number', is defined by

kd = h,/k (2.5)and 8 is the angle between h, and k.

As observed by Lehnert, the character of the decay depends critically on thevalue of 6; if Igl < 1 both roots pl nd p2are real and positive, and p and q decayexponentially without oscillation, while, if 161 > 1, the roots are complex con-jugates and damped oscillating solutions occur. It will be expedient t o consider

FIGURE . The domainsgl,2 ndg3 f wave-number space. Only the half-space cos 8 2 0is indicated. The complete picture is symmetrical about the axis 8 = 0 and about theplane 8 = 4 The locus 6 = 8 s the sphere of radius 28kd touching the k,k,-plane atk = 0.separately the behaviour of the solutions in the following three domains (figure2)of wave-number space

9 1 : 161 < 6 ; B2:61 > 6-1; g3: < 161 < 6-1, ( 2 . 6 )where 6 is any fixed number, 0 < 6 < 1 ; to be specific, and for numerical con-venience, let 6 2-4. At a later stage (see (3.41)below), we shall let 6+ 1 , so thatg3 ill disappear.

(i) The decay of Fourier components ing1Here define a new variable E byThen, from ( 2 . 3 ) , sin[ = 6,p1= hk2cos2 E,

I[[ < &.p2 = hk2sin2&E,


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Suppress ion of turbulence by a magnetic field 577and it is readily verified that the solution of (2 .2)subject to the initial conditionsis

For 151 .g 1, (w 5 and these results take the simpler approximate formp1M h k2 , pZM h k j cos28 , (2.11)and p M p,e-pat, (2.12)

q w @6(e-pat-e-11f)p, N &i


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578 H . K . Moffatt3. Suppression of homogeneous turbulenceFor homogeneous turbulence, the Fourier-Stieltjes representation

( a ) Initial conditions

U = eQSxdP(k,) , h(x, ) = eik%Q(k, ) , (3 .1)s sis appropriate (Batchelor 1953, $2 . 5 ) . The analysis of $ 2 still applies, with thereplacements

The spectrum tensor of the velocity field is defined byp(k, )+dP(k, t ) , q(k, ) +dQ(k, . (3 .2)

Qij(k,t)= (2n)-3 ui(x,t)uj(x+ , t)e4Srdr, (3 .3)sand it is related to dP(k, ) (Batchelor 1953) bydPz(k,t ) d P mdk ,dk ,dk , '.,(k, ) = limdk-0 (3 .4)

where the bar denotes an ensemble average (in view of the spatial homogeneity,the bar in (3 .3)may equally be regarded as a space average). The inverse of (3 . 4 )is

u4(x, )uj(x+ r, ) = Qij(k, )eikJdk, (3 . 5 )K(t )E +U' = - Q4i(k,t)dk. (3 . 6 )

s- '1so that in particular

Let Y,,(k, ) be the magnetic spectrum tensor defined analogously in terms of hand satisfying

d & m , ) t )dk,dk ,dk , *Yij(k, ) = limdk-0 (3 .7)An 'interaction tensor' (see Lehnert 1955) may also be defined, but its propertieswill not be investigated here.

0, i t is clear that the results of 3 2 and therelations (3 .4) and (3 .7) permit us immediately to write down expressions forQ4j(k,) and Y4j(k, ) in terms of Qij(k, ) .We shall be particularly interested inevaluating integral expressions of the type (3 .6) , and, to this end, the form ofQij(k,O)must be specified. Let us suppose that at t = 0 the turbulence is iso-tropic, so that (Batchelor 1953, $3 .4 )

With h(x,0 )E 0, and so dQ(k,0 )

where E,(k) is the initial spectrum function satisfying(3 .9)


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Suppression of turbulence by a magnetic field 579We shall suppose that E,(k) is characterized by a single wave-number k,, so that,on dimensional grounds,

E,(W = aUtkfi1f(k/k,) , ( 3 . 1 0 ) twhere f ( K ) d K = 1.

/omTwo specific choices of the function f (K)will be consideredc K 4Prototype A : f ( K ) =-+ & (3.11)(3 .12)Prototype B: f ( K ) = C K ~ ~ - ~ ; (3 .13)

where, in both cases,C is a constant of order unity determined by (3 . 11 ) .In both(3 .14)ases E,@) N + C ~ E k , 3 - ~ k ~s k+O,

a behaviour that is known to be dynamically self-preserving in the absence ofmagnetic effects (i.e. for t < 0 here) (Batchelor & Proudman 1956) .$In case A,E , ( k ) N $Cu;kjk-B (k9 k,), (3.15)

the behaviour predicted by the Kolmogorov theory for turbulence which has hadtime (prior to the instant t = 0) to attain a state of small-scale statistical equi-librium. The type B spectrum is probably a better representation of turbulenceth at is created explosively a t time t = 0 and that does not have time t o developa Kolmogorov tail before the influence of the magnetic field is experienced.Both prototype spectra have maxima a t a wave-number of order k,. The lengthscale of the energy-containing eddies I,, used in 4 1,may now be identified withthe scale kcl.

There are now two distinct times which characterize the suppression problem,viz. t , andThe ratio of these is t , = ( A k y . (3 .16)

(3 . 17)where S is the Lundquist number, already introduced in 0 1.I f X 4 , then k, < k,, and the bulk of the initial energy is contained in g1(figure3 a ) , indeed in that part ofBl here the approximation IlJ 4 was made.

(3.18)More precisely, 1KAO) = JB1 @,,(k,0)dk = K(O),7 PK,(O)= f J @$,(k,)dk = O ( B 5 ) K ( 0 ) ,

9.(3 .19)

using (3 .14) , valid in g2n this case.t The Reynolds number may occur implicitly in (3.10); owever, as explained in Q 1 aninviscid analysis is appropriate, so that we lose nothing by considering only forms ofE , ( k ) appropriate to the limit R -f 00.$ It has recently been shown (Saffman 1967) that the behaviour E,(k) = O ( k 2 )asb + 0 is also a dynamically self-preserving possibility. The analysis that follows can easilybe adapted to an initial spectrum which behaves in this way.37-2


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580 H . K . Moflattpart of g2or which 161 9 1 (figure 3 b ) . More preciselyIf S 9 1, then k , 9 k,, and the bulk of the initial energy is contained in that

(3 .20)

(3 .21)

FIUURE. The initial energy distribution relative to the domains gl ndg2 the domaing8 s not shown), The distribution is spherically symmetric, and only its variation withwave-number magnitude is indicated. Case ( a ) S < 1; case ( b ) S S- 1, type A spectrum;case (c): S $ 1, type B spectrum. In cases ( b ) and (c ) the dominant contribution to theinitial energy density comes from the shaded regions.Here a distinction has t o be made between the type A and the type B initialspectra. For a type A spectrum, the second term of (3 .21) is dominant and gives

(3 .22)For a type B spectrum, the first term of (3 .21) is dominant and gives

K l ( 0 )= O(S-l)K(O). (3 .23)I n this case we shall seek to describe the decay of energy in the time rangest , < t < t , and t , < t < to . Note that t,/t, = R, < 1, by supposition.

( b ) Kin etic energy decay in glFrom the solution (2 .10)valid in Bl, he replacement (3 .2 )and the relation (3 .4) ,we have immediately the contribution to K ( t ) rom g1

n


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Suppression of turbulence by a magnetic$eld 581Since /cot461 > /tan861 and p 2 < /3 throughout g1,he dominant contributionto K,(t) is given by?

(3 .25)Let ( E , 0,x) be spherical polar co-ordinates in wave-number space, and let,U = cose. Then dk = k2dkd,udX.The X-integration in (3 .25) is trivial and gives afactor 2 ~ .e then change variables from (k, U) to (k, ) ,nd, noting tha t

(3 .26)

For t B t,, the dominant contribution to the [-integral comes from the neighbour-hood of 6 = 0, SO that

where(3 .28)

Now E , ( k ) has a maximum at k = O(k,), so that , providedt 9 ( A k y = t,,

the error function in (3 .28) is unity over the range of k which contributes signi-ficantly to the integral. Hence for t 9 max (t,, t,),

KlV) ( 9 W - w t d / t ) * . (3 .29)Thus, when S< 1, this result is valid when t 9 t,, and, when S > 1, it is valid whent 9 t,.If S 1, and t , g t < ,, then

erf[i(:)*k] M g ( g ) * k for k < (At)-*. (3 .30)The contribution to the integral (3 .28) from the range [(At)-*, 001 is negligible.(since(At)-* > k,), and (3 .28) takes the form

(3 .31)t It will appear (see (3.29)below) that the retained term here is proportional to ( t d / t ) i ,whereas examination of the rejected part of ( 3 . 2 4 ) shows that it is O(t,/t)t, and is alwaysnegligible for t % td .


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582 H . K . Moffatt

td t A *OFIUURE . The decay of K,(t),K&), M l ( t ) ,M,(t), (a)when S < 1, and ( b ) when S %- 1.The scale is logarithmic.where k, = O(ht)b.For a type A spectrum this gives

Kl(t)IKl(O)= O( t , / t ) ~ ( t d < t -4 tA), (3 .32)and for a type B spectrum

Kl(t)IKl(O)= O(1 ) ( t d < t Q t* ) .These results are summarized in figure 4.

( c ) Kinetic energy decay in B2 nd B8The contribution to K(t) romg2s

sinwt a) e-hkat(D).l.l(k,)dk,

(3.33)

(3.34)


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Suppression of turbulence by a magnetic $eldiin the notation of $2(ii). In9,, sinh [I > 1;hence K , ( t ) is of order

e-hkat@iZ(k,) d k=

=1 E o ( k ) 1-$) -hk*fdk.583

(3.35)If S< 1, ( k , < ko) , hen we may use the asymptotic form (3.14) in this integral.Hence K 2 ( t )= 0 [4 C 4 k , 5 jokd4 (1--i,)-Akzfdk). (3.36)For hki t < 1, i.e. t 4 ,, K 2 ( t ) / K 2 ( 0 )O(1). Fort 9 t,,

K,(t) N # 7 u i k o 5 ( h t ) - *= $ C K ( O) X 6 ( t # / t ) & . (3.37)If S 1, ( k , B ko) , hen similarly

(3 .38),(t)/K,(O)= O(1)The behaviour of K,(t) is also indicated in figure 4.

II2 I S&RR,-2

I II II I M2

'd t0FICTJRE. An indication of the dominant contributions to kinetic and magnetic energydensities at different stages of the suppression process.

The contribution to K ( t ) rom g3atisfiesn

using the inequality (2.17).Using the variables (6,k ) his becomes


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5 84 H . K . MoflattHence a repetition of the analysis used above for K,(t) shows that

(3 .40)for t 9 t , if S < 1 , and fort 9 t , ifS $ 1. Hence K 3 ( t )s negligibleat all times. Thesituation S = O(1) would pose special difficulties and is not considered here. Sincethe contribution to the energy from B3 s negligible (and the same applies to t hemagnetic energy) it is legitimate to absorb it in9, nd B2,which are accordingly

9,:151 < 1; B2: gl > 1 . (3 .41)enceforth defined by( d ) Sum mary: the decay of K ( t )

There are three possibilities that must be distinguished when the contributionsK,(t ) and K 2 ( t ) re superposed.1 , then K,(t) is negligible for all t , and the asymptotic behaviour isgiven by equation (3 .2 9 ) .The only significant contribution to the energy comesfrom the neighbourhood of = 0, i.e. 6 = in; .e. the turbulence tends to becomeindependent of the co-ordinate x parallel to h,.(ii) If 1< S < R G ~ ,hen K 2 ( 0 )9 K,(O),but K 2 ( t ) ecreases more rapidly thanK, ( t ) for t 9 t,) and the two contributions become of the same order of magnitudeafter a time of order t, where (by comparing (3 .29) and (3 .3 8 ) )

t, = s a t , = #fita = S*R,t,. (3 .42)For t , < t < t,, K ( t )M K,(t) and is given by (3 .3 8 ) ) while, for t, < t < to,K ( t )M K,(t ) and is given by (3 .2 9 ) . Again for t 9 t,, the turbulence is nearlyz-independent. -

(iii) If S 9R G ~ ,hen K 2 ( t )$ K,(t ) throughout the suppression process. Whent = O(t,), the turbulence is still nearly isotropic (since in g 2 here is no strongtendency to anisotropy-see $ 4 below) and has the character of a randomsuperposition of slowly decaying Alfvh waves.

(i) If S

( e ) Th e u ltimate partition of kinetic energy parallel and perpendicularto the applied jieldThe initial isotropy implies that a t t = 0,- - - _ -U2 = v2 = w2 = *(U"V", (3 .43)where U = (U,v, w). If S 9 R i 2 , hen this isotropy persists throughout the sup-pression process. If S < R z 2 , hen ultimately the turbulence may be described as'nearly two-dimensional', in the sense th at all correlations vary slowly in thex-direction parallel t o h, compared with their variation perpendicular to h,. Thisdoes not imply, however, that G+ 0; indeed, for t 9 max ( E d , t,),

(3 .44)iw2 ISI 3 3 ( k , t ) d k ,_ -U~+ v 2 * [@i,(k, )+@22(k, ) ] k.15%t If non-linear effects become important a t the earlier time t, = N?'to hen the ohange-over will occw only if t , @ t,,, i.e. if S @ R;2r1'6-4?"sae footnote on p. 572).


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Suppression of turbulence by a magnetic$eld 5 85Now from (3 .8) ,

I33(k,0)~ ~ ( l - c o s 2 1 9 ) , (3 .45)Since the dominant contribution to the integrals (3 .44) comes from the neigh-bourhood of 8 = in, he contribution from the cos28 erms is negligible, and so

w2 w U2+v2, (3 .46)in contrast to (3 . 43 ) .The energy is therefore, to some extent, channelled duringsuppression into the component parallel to the field.

ultimately - _ _

(f) Magnetic energy development and decayThe magnetic energy may be treated in the same way as the kinetic energy. Itwill be enough to indicate here the essential differences and conclusions. Themagnetic energy rises from zero (by virtue of the special assumption h(x, )= 0 )t o its maximum value in a time of order t,. For t $ t ,

MZ(t)= K2( t )= O[K,(O) th/t)*I (3 .47)i.e. in g2 here is approximate equipartition of energy; this follows essentiallyfrom the behaviour of the Fourier components (2 .15) for 161 $ 1 . In g1he maincontribution to Ml( t )again comes from the neighbourhood of 8 = Bn,where, asobserved after equation (2 .13) , the magnetic energy in each Fourier componentis a factor ( kd / ko ) 2os28 smaller than the kinetic energy. The integral overg1 fYi t (k , ) for t B t , is of order

(3 .48)and for t $ max ( t d ,,) thisrgives

J4( t ) = O[S2K(O)( td / t )4I , (3 .49)which may be compared with (3 . 29 ) .Note tha t

M l ( t ) / K l ( t )= O[fJ2(td/t) l O(t, /t) . (3 .50)If S @ 1, then M ( t )wM l ( t )a t all times. If X $ 1, then, as for the kinetic energy,M 2 ( t )$ M l ( t )up to a time of order tm ,where, by comparing (3 .47) and (3 .49) ,

t, = X3t&If t , < to , .e. if 1 < S @ 2 ~ ~ )hen (3 .51)

(3 .52)If S $ R i l , however, then tm B to , so that M ( t ) N M 2 ( t ) hroughout the sup-pression process.? The various possibilities are summarized in figure 5. Note tha tt In this case tm g t , 0 g R;Yl(s-2Y);of . footnote on p. 584.


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586 H . K . Moffattif R&1


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Suppression of turbulence by a m agnetic i e l d 587Some numerical computations that are relevant to the present study have been

made by Deissler (1963) . The computations were made for several values of theratio vlh , and only the results for v / h< 1 need be considered here. The main con-clusions for this case were (i) hat the magnetic energy was negligibly small com-pared with the kinetic energy for as long as the computations were continued;(ii) hat 3w2/u2ncreases from 1 to $ during the process of suppression. Both con-clusions are borne out by the analysis of this paper in the caseS < 1 (see $3 e )and (f)).Perusal of Deisslers paper shows that his computations are in factrelevant only to this case; the asymptotic form (3.14) s adopted as an initialcondition for all k and this is legitimate only if the bulk of the initial energyis in g1,.e. only if S

When t = O(t,) and later, non-linear terms in the equation of motion will tendto redistribute energy in wave-number space, and thus lead indirectly to a changein the laws of energy decay. The prevailing tendency in ordinary turbulence isfor non-linear terms to transfer energy towards the viscous sink a t high wave-numbers. In the problem under consideration, in addition to this effect, we shouldexpect a transfer of energy from the neighbourhood of p = 0 (where there is noohmic:dissipation) towards the region p = O(l),where ohmic dissipation is strong.If S is small this will undoubtedly accelerate the decay of energy in Bl;.e. K(t)will decrease more rapidly than ( td / t )* . t is clear from a consideration of (4.1)that the non-linear term will first become significant in the important neighbour-hood of p = 0,where the only other term contributing to spectral change is small.Relative to the magnetic force, the inertia force is most important in this region,and its influence is therefore likely to be felt a t an earlier stage than the originalcrude estimate would suggest (see footnote on p. 2 ) .

If 8 is sufficiently large, non-linear effects may be less important. The Alfvdnwaves in B2 re of small amplitude in the sense ($)!L, (@)*< h, provided S 9 R(from a consideration of the definition of these two numbers), and interaction ofdecaying Alfvdn waves may do no more than slowly alter the spectral shape asdescribed by the linear theory (equation (4.3)). owever, there may be a signi-ficant transfer of energy from B2oBl,n the absence of a detailed model for theprocess of non-linear transfer, it is impossible to say in which domain of k-spacethe dominant contribution to the energy ultimately resides.

- -

1.

Appendix. Relation with previous work on the final period of decayAs mentioned in the introduction, the linearized MHD equations were studied

by Lehnert (1955) in the context of the final period of decay, th at is, under cir-cumstances wherein the velocity and magnetic fluctuations are sufficiently weakfor non-linear effects to be permanently negligible; sufficient conditions for thisto be the case areR and Rm eing based on the r.m.s. velocity and on I, = kcl, he length scale ofthe energy-containing eddies at some initial instant marking the beginning ofthe final period. The viscous term vV2u must then be included in the equation ofmotion ( 1 . 7 ) ; otherwise the equations studied in this paper are unaltered.

(A 1)< 1, Rm < 1,


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5 8 8 H . K . MoffuttSeveral authors have continued the study of Lehnerts equations and it may

be as well to review briefly their conclusions, and to set them in perspective withthe present paper. The question of the asymptotic law of decay appears to havebeen first examined by Alexandrou (1963),who demonstrated that, when h = v ,the kinetic and magnetic energy densities were asymptotically (i.e. for t -+ 00)equal, and proportional to t-8 as for non-magnetic turbulence (Batchelor 1953,$5.4).The result depends critically on the behaviour of the spectrum tensors@.,,(k, ) and Yij(k,0) near k = 0, since essentially it is the Fourier componentsof largest scale which survive longest. The nature of the spectral tensors in thisneighbourhood is determined by non-linear interactions during the completehistory of the turbulence prior to the final period of decay. This aspect of theproblem was examined thoroughly by Alexandrou following the assumption andmethod of Batchelor & Proudman (1956).The essential property of the spectraltensors that he obtained for axisymmetric turbulence was

@&, 0 ) k2f0(P, Y.,.,(k,0 ) k290(P2) (k + 01, (A 2)where fo and go are functions which depend on the previous (i.e. t c 0) non-linearhistory of the turbu1ence.t It is then integrals of the form1 .,.,(k,) cos2 h0kpt ) -A k Zt d k ,which asymptotically give contributions proportional to t-8. Alexandrou arguedfurther that the t-8 law was likely to be asymptotically valid if h + v provided

The decay of turbulence during the final period in the weakly conducting casehas been further considered by Lecocq (1962),Eliseev (1965a, ) and Nihoul(1965, 1966).Lecocq obtained the particular form of Lehnerts spectral decayresults appropriate to the situation v < A. Eliseev, and Nihoul (1965), xaminedthe decay of kinetic energy density during the final period, and independentlyobtained the result

h / v = O(1).

K ( t )N t4 ( t / t d + ~ ~ ) . (A 3)I n essence, in these treatments, only the contribution to K ( t ) rom the domaing1 f k-space was considered. However, the contribution from g2n fact fallsoffmore slowly than that fromg1,nd Nihoul(l966)has now given the modifiedresult

It has not previously been recognized that the decay of K ( t ) n the final periodmust depend significantly on the value of the initial Lundquist number S as wellas on the ratio h/v. It will now be shown that Nihouls result (A 4) is essentiallycorrect if S < ( v /h )*< 1 (see (A 13) elow), but tha t it requires further modifica-tion if S 9 (v /h)+.

t In view of Saffmans (1967) work, it is now evident that this is perhaps not the mostgeneral possibility.


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Suppression of turbulence by a magnetic$eld 589In order to describe the final period of decay, the analysis of $92-3 requiresmodification only through the inclusion of the viscous term vV2u n the equationof motion ( 1 . 7 ) .The modification in the analysis of $ 2 is trivial. The definition

( 2 . 5 )must be replaced byand ( 2 . 3 )becomes k , = ho/lA-vl , (A 5 )

(A 6)1,2= $(A+Y )k2 $(A- )k2(1- 2) i .I f h = v , then k , = 00, so that the whole k-space (except the singular plane0 = &r) lies ing2.n this case the analysis of Q 3 ( c ) again gives

K,(t)/K,(O) f ( t h / t ) P (t B max(t,,t,)), (A 7)consistent with Alexandrou ( 1963 ) .This result is insensitive to the choice of thefunctions fo and goappearing in the initial conditions (A2) apart from a constantof proportionality of order unity; all that is required is the factor k2 in theseconditions.Suppose now that h $ v (results for the case h < v may be inferred by virtueof the symmetry of the governing equations). Then E , w ho/has in $ 2 , and thedomains 93, and g2 re as before. There are now three time scales that must beconsidered:

t , = h/h& t, = (Akt) - l , t, = ( vk t ) - l , (A 8)and, by assumption, t, $ t,.

It is not difficult to see that the decay in 5B2is not materially changed when vis non-zero; the amplitude of each Fourier component decays approximately ase -3 ,k2t , and the decay of K 2 ( t )s still given by (A7). However viscous effects doultimately become important in9,;n the region < 1, the roots (A 6) become

p1w hk2, p2 w h k ~ c o s 2 0 + v k 2 .This leads to/.K,(t) w @i,(k,0)exp{-2(p2t/td+vk2t)}dk.91

Clearly for t < t, the viscous factor is unimportant. The factor exp (- tp2 / td )contributes a factor ( td/t)* o K,(t) for t $ max (td,t,) (just as in $ 3 ( c ) )and thefactor e-2vk2t contributes a factor (t,/t)t to K,(t) for t $ t,. (Here we consider onlya type B initial spectrum, this being the more appropriate for a final period ofdecay calculation.) Hence if t , < t,, then

K,(t) w K , ( ~ ) ( t , / t ) * or max (t,, t,) < t < t,, (A 10)1~,(o)(t , / t ) ic( t , / t )&or t B t,,and, if t , 9 t,,(A 1 1 )I,(t) w K,(o)(t,/t)Q for t, < t < t,,w K,(O)(t,/t)S t,/t)& for t B t,.

I n these results, and in those that follow, constants of order unity are omitted.four distinct possibilities summarized in figure 6.

When the contributions from 9 , andg2 re superposed, there are essentially


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5 9 0 H. . Moffatt(i) S 4 u/h))4 , i.e. t , < t, 4 ,. In this case, K,(t)dominates until a time

t , = ( h / Y ) 5 t d , (A 12)(A 13)(6, e t 4 d ) ,and

consistent with Nihoul's (1967) result (A 3) .time t,; however, (A 13) clearly requires modification(ii) (u/h)+ S < 1, i.e. t , < t , < t,. In this caseK,(t) again dominates until the

(A 14)( t d e t < t,),

I I I I - I ' -

td t A t V tFIUURX.Decay of R,(t)and K,(t) in the h a 1 period for different orders of magnitude ofthe parameter S. The scale is logarithmic.


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Suppression of turbulence b y a magnetic$eld 59 1(iii) 1 < S < ( A / v ) ~ .n this case K,(t) dominates for t < toand for t > tf, where

tc = Sit,, tf = S-2(A/v )4 tv , (A 15)

( t h Q t 4 c ) ,K(t ) s-1(t ,/ t)i (t , Q t e t,),(t< t 4 f),

but K,(t) dominates for tc< t tf. Hence(A 16)1t A / t ) & ( t9 f).( t h / t ) QK(O) fi : s-l(tJt)*(t,/t)Q

(iv) 8; ( A / v ) 2> 1. In this case K,(t) dominates throughout the final periodof decay, and

Is = O(V/h)+ h / v < 1L0 S

FIGURE. Schematic division of the (8 ,h/v)-plane into the four regions (for h/v S 1) inwhich distinct patterns of decay occur. The numbers (i)-(iv) correspond to the numbersused in the text . The region h/v < 1 could be similarly divided.The reason that the cases (ii), (iii) and (iv) are not embraced by Nihoulsanalysis is that , like Deissler, he adopts initial conditions of the form (A 2) and

applies them as though they are valid for all k ; his is legitimate only in the limitk,,+co, i.e. S+O, so that only the case (i), in whichS is smaller than any othersmall numbers appearing in the analysis, is adequately described.The regions of validity of the results (A 13)-(A 17) in the plane of the variables(S,A / v ) is indicated in figure 7 . In the limit A/v+co, for fixedS, only the regions(ii) and (iii) survive, and t,+co. The spectacular variability of the results in the

different regions in this trivial type of linear turbulence may serve as a warningof the complexity of behaviour that may be anticipated under fully developednon-linear conditions.


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592 H . K. MoffattR E F E R E N C E S

ALEXANDROU,. 1963 Ph.D. Thesis, Kings College, London.BATCHELOR,. K. 1953 Homogeneous Turbulence. Cambridge University Press.BATCHELOR,. K. & PROUDMAN,. 1956 P h i l . Tr an s . A 248, 369.DEISSLER, . B. 1963 Phys . F lu ids 6 , 1250.ELISEEV, . V. 1965la So viet Ph ysic s, Doklady 10,239.ELISEEV, . V. 1965b J . Appl. M a t h . M ech. (PMM) 29 , 1133.GOLITSYN,. S. 1960 Soviet Physics, Dolclady 5, 536.LECOCQ,. 1962 C. R. Acad. S c . P a r i s , 254, 3633.LEHNERT, . 1955 Quart. Appl. M a t h . 12, 321.LIEPMANN,. W. 1952 2.Appl. M a t h . P h y s . 3, 321.MOEFATT,. K. 1962 Mdcan igue de la Turbulence, p. 395. Editions du C.N.R.S. no. 108.Paris.NESTLERODE,. A. & LUMLEY, . L. 1 96 3 P h ys . F lu i d s 6, 1260.NIHOUL,.C . J. 1965 Physica 31, 141.NIHOUL,.C . J. 1966 P hys . Flu ids 9, 2370,SAFFMAN,. G. 1967 J . P l u i d M ech . 27, 581.

H.K. Moffatt- On the suppression of turbulence by a uniform magnetic field

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